AI 中文总结
该研究针对超空间共变环的希尔伯特级数系数，给出显式正组合解释与生成函数，特化时证明Sagan–Swanson 2024年的回文猜想，还得到任意m≥1时u=-q^m特化的闭式表达式。
AI 中文摘要
确定超空间共变环的希尔伯特级数的系数由钩形分划索引。我们给出这些系数的显式正组合解释，以及它们的若干生成函数。将该希尔伯特级数在 $u=-q^2$ 处特化时，我们证明其系数是二项式系数的差。由此，这证明了Sagan–Swanson（2024）的猜想：这些系数在符号上是回文的。更一般地，对每个 $m \geq 1$，我们得到 $u = -q^m$ 特化的闭式表达式。
英文摘要
The coefficients that determine the Hilbert series of the superspace coinvariant ring are indexed by hook-shaped partitions. We give a manifestly positive combinatorial interpretation of these coefficients, together with several generating functions for them. Specializing this Hilbert series at $u=-q^2$, we show that its coefficients are differences of binomial coefficients. Consequently, this proves a conjecture of Sagan--Swanson (2024) that these coefficients are palindromic up to sign. More generally, for every $m \geq 1$ we obtain closed-form expressions for the $u = -q^m$ specialization.
Comments22 pages, 1 figure. Comments are welcome
